We still need to convince ourselves of some basic facts about the previous lecture, for example is the map A \to A *_C B injective?

Example: Cut the sphere S^2 along the equator. Then the diagram we have is

Definition: If G=A *_C B or G = A*_C we say that G splits over C, and we call C the edge group. If G=A*_C or G = A *_C B and C is not A or B in the latter case, then we say that G splits non-trivially.

Definition: Let \Gamma be a connected graph (i.e. a 1-dimensional CW-complex). For each vertex v \in V(\Gamma) (resp. edge e \in E(\Gamma)) let G_v (resp. G_e) be a group. If v_\pm are vertices adjoining an edge e then let \partial_\pm^e : G_e \to G_{v_\pm} be an injective homomorphism. This data determines a graph of groups \mathcal G.

We say that \mathcal G has:

  • underlying graph \Gamma
  • vertex groups \{ G_v \}
  • edge groups \{ G_e \}
  • edge maps \{ \partial_\pm^e \}

Similarly, we have:

Definition: Let \Xi be a connected graph. For each vertex v \in V(\Xi) (resp. e \in E(\Xi)) let X_v (resp. X_e) be a connected CW-complex. If v_\pm adjoin e let \partial_\pm^e : X_e \to X_{v_\pm} be \pi_1-injective continuous maps. This data determines a graph of spaces \mathcal X. It has underlying graph \Xi, vertex spaces X_v, edge spaces X_e, etc. The graph of spaces \mathcal X determines a space as follows: define

\displaystyle X_{\mathcal X} = \left(\bigsqcup_{v \in V(\Xi)} X_v \sqcup \bigsqcup_{e\in E(\Xi)} (X_e \times [-1,+1])\right) / \sim,

where (x,\pm 1) \sim \partial_\pm^e(x) for x \in X_e. We say that \mathcal{X} is a graph-of-spaces structure (or decomposition) for X_\mathcal{X}.

Remark: There is a natural map X_\mathcal{X} \to \Xi (by collapsing all the edge and vertex spaces).

Given any graph of groups \mathcal G we can construct a graph of spaces \mathcal X with underlying graph \Gamma by assigning X_v = K(G_v,1), X_e = K(G_e,1) and realizing the edge maps as continuous maps X_e \to X_{v_\pm}. We write X_\mathcal G for X_\mathcal X. This is well-defined up to homotopy equivalence.

Definition: The fundamental group of \mathcal G is just \pi_1(\mathcal G) = \pi_1(X_\mathcal{G}).

Examples:

  • If then \pi_1 (G) = A *_C B
  • If One vertex labeled A, one edge labeled C then \pi_1 (G) = A *_C
  • Let \mathcal C \subset \Sigma be an embedded multicurve (disjoint union of circles) inside a surface. Cutting along \mathcal C decomposes \Sigma into a graph of spaces and \pi_1 (\Sigma) into a graph of groups.

Note: The edge maps of X_\mathcal{G} are only defined up to free (i.e. unbased) homotopy. Translated to \mathcal G, this means that only the conjugacy class of \partial_\pm in G_{v_\pm} matters.

Remark: The map X_\mathcal G \to \Gamma induces a surjection \pi_1(\mathcal G) \to \pi_1(\Gamma).

Here’s a way to construct a graph of groups. Let’s suppose G acts on a tree T without edge inversions (we can do this by subdividing edges if necessary). Let Y = \widetilde{K(G,1)}. The group G acts diagonally on T \times Y. The quotient X = G\setminus (T \times Y) has a structure of a graph of spaces. The underlying graph is \Gamma = G\setminus T and there is a natural map X \to \Gamma.

Let v \in V(\Gamma) be a vertex below \tilde v \in V(T). The preimage of v is just v \times (G_{\tilde v} \setminus Y) where G_{\tilde v} is the stabilizer of \tilde v. Similarly, for e \in E(\Gamma) below \tilde e \in E(T), the preimage of e is e \times (G_{\tilde e} \setminus Y).

If \tilde e adjoins \tilde v then G_{\tilde e} \subset G_{\tilde v} so the edge map G_{\tilde e} \setminus Y \to G_{\tilde v} \setminus Y is a covering map and therefore \pi_1-injective. We have defined a graph of spaces \mathcal X and \pi_1(X_\mathcal X) = G since T \times Y is simply connected.

Applying \pi_1 to everything, we have a graph of groups \mathcal G. Its underlying graph is \Gamma. Its vertex groups are the vertex stabilizers of T, its edge groups are the edge stabilizers, and the edge maps are the inclusions. Also, \pi_1(\mathcal G) = G.

Question for next time: Does every graph of groups arise in this way?

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